It was only the third time in his career that world number one Woods had missed the cut.
Woods followed his opening 69 at the TPC at Four Seasons Resort Las Colinas with a 72 at the Cottonwood Valley Golf Club, the other course being used this week, as he failed to beat the halfway guillotine by one stroke with a one-over total of 141.
The last time he missed the cut was at the 1998 Pebble Beach Pro-Am.
"It's been a nice little, how many years, seven years?", the 29-year-old American told reporters. "That's not too bad.
"It's never a relief when you miss a cut. I've missed three of them so far in my career and none of them feel very good, especially when you bogey the last hole."
Woods was level-par for the event when he bunkered his approach at the 18th. He splashed out to 15 feet but failed to sink his putt for a regulation four.
"It's disappointing any time you miss the cut because you're here for four rounds to try to compete to win and now I don't get that opportunity to win the golf tournament.
"I just had a tough day. It's just one of those days."
Ironically, it was Nelson's 54-year-old PGA Tour record of 113 consecutive cuts that Woods eclipsed in November 2003.
At the top of the leaderboard on Friday, O'Hair fired a five-under 65 and Wetterich recorded a 67 as the two unheralded Americans finished nine-under on 131.
O'Hair, 22, who has one top-15 finish to his credit from 11
Wetterich, 31, has missed the cut in his last three outings.
Ted Purdy was alone in third place after compiling a 67 for 132, one in front of Billy Mayfair (63).
World number four Phil Mickelson carded a 66 and was among a group of nine players on five-under 135.
Fiji's Vijay Singh, who can reclaim top spot in the rankings with a top-three finish, was also on five-under after the world number two registered a 67.
Playing in his second event since capturing his fourth U.S. Masters title last month, Woods struggled to shake the rust off his game.
He was two-under for the tournament through eight holes of his second round before slumping with bogeys on nine, 13 and 15.
A birdie on the par-five 16th gave him hope but that was extinguished at the final hole.
"It was a very frustrating round because I couldn't quite find where I needed to put the club in the right position to actually make a golf swing," said Woods, who has won three times this season.
World number five Retief Goosen of South Africa also missed the cut by a stroke after a 71 for 141.